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V6-11

Mathematics Grade 9W3 - Review

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Copyright © 2011, by Alberta Distance Learning Centre, 4601-63 Avenue, Barrhead, Alberta, Canada, T7N 1P4. Additional copies may be obtained from Alberta Distance Learning Centre.

No part of this courseware may be reproduced or transmitted in any form, electronic or mechanical, including photocopying (unless otherwise indicated), recording, or any information storage and retrieval system, without the written permission of Alberta Distance Learning Centre.

Every effort has been made both to provide proper acknowledgement of the original source and to comply with copyright law. If cases are identified where this effort has been unsuccessful, please notify Alberta Distance Learning Centre so that appropriate corrective action can be taken.

IT IS STRICTLY PROHIBITED TO COPY ANY PART OF THESE MATERIALS UNDER THE TERMS OF A LICENCE FROM A COLLECTIVE OR A LICENSING BODY.

Mathematics Grade 9Version 6Preview/Review W3 - ReviewISBN: 978-1-927090-00-8

Publisher: Alberta Distance Learning CentreWritten by: Lenee FyfeReviewed by: Danielle Winter

Project Coordinator: Danielle WinterPreview/Review Publishing Coordinating Team: Julie Reschke

Alberta Distance Learning Centre has an Internet site that you may find useful. The address is as follows: http://www.adlc.ca

The use of the Internet is optional. Exploring the electronic information superhighway can be educational and entertaining. However, be aware that these computer networks are not censored. Students may unintentionally or purposely find articles on the Internet that may be offensive or inappropriate. As well, the sources of information are not always cited and the content may not be accurate. Therefore, students may wish to confirm facts with a second source.

Materials Required

PencilPaperCalculatorTracing PaperGrid Paper

Important Concepts of Grade 9 Mathematics

No Textbook Required

This is a stand-alone course.

W1 - Lesson 1 .............................................................................................. PowersW1 - Lesson 2 .........................................................................................ExponentsW1 - Lesson 3 ............................................................................Rational NumbersW1 - Lesson 4 ......................................................................... Order of OperationsW1 - Lesson 5 .................................................Square Roots of Rational NumbersW1 - ReviewW1 - Quiz

W2 - Lesson 6 ..............................................................Graphing Linear RelationsW2 - Lesson 7 ................................................................. Solving Linear RelationsW2 - Lesson 8 .......................................................................... Linear InequalitiesW2 - Lesson 9 ...................................................................................... PolynomialsW2 - Lesson 10 ............................................................Surface Area of 3D ObjectsW2 - ReviewW2 - Quiz

W3 - Lesson 11 .......................................................................Properties of CirclesW3 - Lesson 12 ....................................................... Polygons and Scale DiagramsW3 - Lesson 13 .....................................................................Rotational SymmetryW3 - Lesson 14 .........................................................................Representing DataW3 - Lesson 15 ......................................................................................ProbabilityW3 - ReviewW3 - Quiz

Preview/Review Conceptsfor

Grade Nine Mathematics

W3 – Review

Preview/Review Concepts W3 - Review Mathematics Grade 9

Developed by Alberta Distance Learning Centre ..................................................................................................... 1

W3 – Review

Materials required:

• Paper, Pencil, Calculator, Tracing Paper, Grid paper

Part 1: Properties of Circles

Properties related to angles in circles can be used to solve problems. To solve problems, properties of a circle need to be defined.

A chord is a line segment that joins two points of a circle.

C

B

O

A

P

OC = radiusAB = chord

The chord AB had end points that fall on the circle.

A tangent is a line intersecting only one point on the circle; it is perpendicular to the radius.

Tangent

Point of Tangency

Radius

Mathematics Grade 9 Preview/Review Concepts W3 - Review

2 .................................................................................................... Developed by Alberta Distance Learning Centre

“x” is the inscribed angle. It is formed by the two chords AB and BC.

“y” is the central angle.

The central angle is twice the value of the inscribed angle.

Both angles are subtended by the same arc of the circle.

C

A

yO

Bx

A

B

Angle A = Angle B

Intercepted Arc

The inscribed angles A and B are congruent.

Angle A and Angle B are both 35o.

The measure of the central angle is twice the value of an inscribed angle subtended by the same arc. The central angle would be 70o.

The perpendicular bisector from the centre of a circle to a chord will bisect the cord.

F

DE

C

A

G

This is the chord. It is perpendicular because it forms a 90 degree angle with the line passing through the centre.



Example 1

Determine the measure of each angle.

a. ∠B = 24o since ∠A = ∠B

b. ∠C = 48o since the measure of the central angle

is equal to twice the measure of the inscribed angle subtended by the same arc.

Example 2

What is the length of chord CD?

To determine the length of CD, use the Pythagorean Theorem since triangle ACB is a right angled triangle.

a b c

CB

CB

CBCBCD

2 2 2

2 2 2

2

10 26

576

5762424 24 48

+ =+ =

=

=== + =So cm

A

B

Angle A = Angle B

Intercepted Arc

C

E

D

24o

A

DCB

B

26 cm 10 cm



Example 3

The center of the circle is O.

AO = 10 cm

DB = 4 cm

a. What is the length of OC? AO = CO = 10 cm

b. What is the length of OD? 10 – 4 = 6 cm

c. What is the length of CD? 62+ a2 = 102

a = 8 cm

Example 4

What is the measurement of the unknown length?

To solve, use the Pythagorean Theorem to find the unknown side.

a b c

c

c

c

cc

2 2 2

2 2 2

2

2

15 20

225 400

625

62525

+ =+ =+ =

=

== m

A D

C

BO

?

15 cm20 cm



Practice Questions

1. What is the measure of ∠A?

C

A

?

60oB

2. What is the length of chord CD rounded to the nearest decimal?

A

DCB

B

30 cm 5 cm



3. The center of the circle is O.

AO = 25 cm DB = 12 cm

a. What is the length of OC?

b. What is the length of OD?

c. What is the length of CD?

4. Find the missing length.

A D

C

BO

?

200 m

100 m



Part 2: Properties of Polygons

Similar polygons may be a different size or orientation, but they must be the same shape. To check if polygons are similar, see if they have corresponding sides and corresponding angles.

To solve for any unknown side in similar polygons, use scale factor or set up a proportion. If the scale factor is greater than 1, the image has been enlarged. If the scale factor is less than 1, the image has been reduced.

Example 1

Use grid paper to draw the design with a scale factor of 0.5.

L = 4 × 0.5 = 2W = 6 × 0.5 = 3

The reduced image should have new dimensions of 2 × 3. The scale factor is 0.5 units. This is a reduction.



Example 2

The following triangles below are similar. Determine the length of the missing side m.

64

Triangle X

∠C 70o∠B 85o

∠A 25o

24Side m

8

Triangle Y

∠F 70o∠E 85o

∠D 25o

34

To find the missing side, compare the corresponding sides to determine the scale factor. Once the scale factor is known, solve for the missing side.

ABDE

m=4

BCEF

=

=

648

8

ACDF

=

=

243

8

The scale factor is 8. This is an enlargement.

Using algebra, m

m

m

48

44 8 4

32

=

× = ×

= o



Example 3

The following rectangles below are similar. Determine the length of the missing side x.

49 cm

Side x

6 cm

4.5 cm

Rectangle YRectangle X

Since the rectangles are similar, the side lengths will be proportional. Set up a proportion to solve for the unknown side.

649

4 5

6 220 566

220 56

36 75

=

=

=

=

.

..

.

xxx

x cm

The missing side length on rectangle x is 36.75 cm.


10 .................................................................................................. Developed by Alberta Distance Learning Centre

Practice Questions

1. Solve the following proportions. Round to the nearest tenth.

a. 56 450

= x b.

220

3 5= .x

c. 1 4

320x=

2. The average height of a Holstein cow is 175 cm. What was the scale factor used to create this image of the cow?

25 cm


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3. The following triangles are similar. Find the missing side and angles.

35 mm

Triangle O

∠B 45o∠A 75o

∠C o

40 mm

Triangle P

C

A B

F

D E14 mm

x16 mm

∠E 45o∠D 75o

∠F o

12 mm



4. The following polygons are similar. Find the length of the missing side y.

P

Q

RS

6 cm

27 cm

45 cm18 cm

9 cm

15 cm

2 cmLM

NO

Polygon P Polygon O

Side y



Part 3: Rotational Symmetry

Rotational symmetry occurs when an image can be turned about its centre of rotation so that it fits onto its shape more than once in a complete 360 degree turn.

Polygons, images and designs can have rotational symmetry and line symmetry.

Lines of symmetry occur when a shape or image can be divided into identical halves. These identical halves are mirror images of each other and were created by the line of symmetry.

Example 1

How many lines of symmetry does the image have?

This shape has 4 lines of symmetry.



Example 2

a. What is the order of rotation?

b. What is the angle of rotation?

c. Where is the centre of rotation?

d. How many lines of symmetry does the shape have?

4

3604

90o

o==

The centre of the shape.

0



Practice Questions

1. How many lines of symmetry does each of the following shapes have? a. b.

2. Consider the following shape.

a. What is the order of rotation?

b. What is the angle of rotation?

c. Where is the centre of rotation?

d. How many lines of symmetry does the shape have?



3. a. Draw the image with a scale factor of 4.

b. Is it an enlargement or a reduction?



4. a. Draw the image with a scale factor of 0.5.

b. Is it an enlargement or a reduction?



Part 4: Data Analysis

Survey questions should be worded so that they are free from any influencing factors that may impact the survey positively or negatively. Factors that can influence a survey include:

Bias Cultural Sensitivity Use of Language Ethics Time Timing Cost Privacy

Example 1

The first 40 people to enter a Home and Garden Trade Show were asked if they would support an increase to the entrance fee for the show, from $5.00 to $6.00.

Here, the influencing factor is the wording of the question. The words increasing the entrance fee could negatively influence people. No one wants to pay more money.

Populations and Samples

To conduct a survey, either a population or a sample can participate. A population refers to the whole group of individuals being studied. However, sometimes this method of data collection is not always practical or cost effective. If the population is too large, using this method is not practical.

Ensure that the sample is random. This means that everyone in the population has an equal chance of being chosen. A biased sample will influence the survey. A bias will impact results by favouring one outcome.



Example 2

A survey is planned to find out what Albertan Grade 12 students plan to do after high school. It would be impossible to survey all grade 12 students, so a sample can be used instead. A sample is any part of the population.

Using Data to Draw Conclusions

Conclusions can be drawn from theoretical and experimental probability. Probability can also be very helpful when making a decision.

Example 3

A researcher wishes to determine the most common eye colour for teenagers in the area. 2302 students from three local high schools were asked to identify their eye colour.

Eye Color TotalBrown 1656Blue 483Hazel 115Green 22Other 26

Using these results, predict how many students out of 5000 would have blue eyes. Round the answer to the nearest percent.

4832302

20 98

21

21 50000 21 50001050

=

=

= ×=

. %

%.

of

Based on the results of the survey, we can predict that 1050 students out of 5000 would have blue eyes.



Practice Questions

1. For each situation, decide if a survey should be conducted using the population or a sample.

a. A survey of teachers, parents, students and administrators to see whether the library in the school needs to be updated.

b. A survey of customers to determine most popular soda flavour.

2. Ask 10 people selected at random in a parking lot if they like to drive cars. What are the influencing factors?

3. 2302 students were asked to identify their eye colour to determine the most common eye colour in high schools.

Eye Color TotalBrown 1656Blue 483Hazel 115Green 22Other 26

a. Based on the above results, predict how many people out of 7500 would have brown eyes.

b. What assumptions are made?

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